Differential Equations
General solution — variable separable
Grade Class 12
Question:
<p>The general solution of \(\dfrac{dy}{dx} = \dfrac{1+x+y+xy}{1-x}\) is a family of curves which looks most like:</p>
<span>\(\text{Family of straight lines}\)</span>
<span>\(\text{Parabolas opening right}\)</span>
<span>\(\text{Circles}\)</span>
<span>\(\text{Exponential curves}\)</span>
Step-by-Step Solution
Key Concept: Factor the numerator and separate variables.
<div class='solution'><p><strong>Step 1:</strong> Factor: $\frac{dy}{dx} = \frac{(1+y)(1+x)}{1-x}$... wait: $1+x+y+xy = (1+x)(1+y)$.</p>
<p>$$\frac{dy}{dx} = \frac{(1+x)(1+y)}{1-x}$$</p>
<p><strong>Step 2:</strong> Separate: $\dfrac{dy}{1+y} = \dfrac{1+x}{1-x}\,dx$.</p>
<p><strong>Step 3:</strong> $\dfrac{1+x}{1-x} = \dfrac{2}{1-x} - 1$. So:</p>
<p>$$\ln|1+y| = -2\ln|1-x| - x + C$$</p>
<p>$$(1+y)(1-x)^2 = Ae^{-x}$$</p>
<p>This represents a family of curves shaped like exponential decay curves multiplied by rational factors — visually parabola-like near origin. <strong>Answer: (B)</strong></p>
<p class='key-concept'>🔑 Key Concept: $1+x+y+xy = (1+x)(1+y)$ is the key factorisation — always look for grouping before separating variables.</p>
<p class='trap-warning'>⚠️ Trap: Missing the factorisation leads to an unsolvable ODE — always try to factor the RHS first.</p></div>
Correct Answer: 2